Talk:Triangulation (geometry)

geometry

What is advanced geometry?

Common face?

The first definition is

A triangulation T of $\mathbb {R} ^{n+1}$ is a subdivision of $\mathbb {R} ^{n+1}$ into (n + 1)-dimensional simplices such that any two simplices in T intersect in a common face or not at all ...

This forbids two simplices to intersect in a common edge, or a common vertex. With this restriction, very few things can be triangulated. For example, a square can be triangulated, but larger polygons cannot. Is this really what was intended? Or have I misunderstood something? Maproom (talk) 11:18, 10 April 2012 (UTC)

In this context, vertices are 0-dimensional faces, edges are 1-dimensional faces, triangles are 2-dimensional faces, tetrahedra are 3-dimensional faces, etc. The faces that have dimension one less than the dimension of the simplex are called facets. Your complaint would be accurate if it said that they intersect in a common facet, but "face" is more inclusive. —David Eppstein (talk) 15:30, 10 April 2012 (UTC)

Thank you for your explanation. That makes sense. I wonder how many readers will misunderstand the definition as I did – but I shan't worry about it. Maproom (talk) 18:22, 10 April 2012 (UTC)